Differential Geometry and its Applications最新文献

Let M be a smooth manifold and $F$ a Morse-Bott foliation with a compact critical manifold $\Sigma \subset M$. Denote by $D\left(F\right)$ the group of diffeomorphisms of M leaving invariant each leaf of $F$. Under certain assumptions on $F$ it is shown that the computation of the homotopy type of $D\left(F\right)$ reduces to three rather independent groups: the group of diffeomorphisms of Σ, the group of vector bundle automorphisms of some regular neighborhood of Σ, and the subgroup of $D\left(F\right)$ consisting of diffeomorphisms fixed near Σ. Examples of computations of homotopy types of groups $D\left(F\right)$ for such foliations are also presented.

The purpose of this paper is to clarify and extend the result of K. Okumura in [7] concerning hypersurfaces in the non-flat complex space forms $C{P}^n$ and $C{H}^n$ whose *-Ricci tensor is $D$-recurrent.

In this article, we study the Zermelo navigation problem with and without obstacles from a theoretical point of view and look towards some computational aspects. More intuitively, this navigation model is in fact an optimal control problem with continuous inequality constraints. We first aim to study the structure of these optimal trajectories using the geometric aspects of the problem. More precisely, we find the time-optimal trajectories and characterize them as geodesics of Randers metrics away from the danger zone and geodesics of (not necessarily Randers) Finsler metrics where they touch the boundary of the danger zone. We demonstrate some of the important behavior of these trajectories by examples. In particular, we will calculate these trajectories precisely for the critical case of an infinitesimal homothety which, in the language of optimal control problems, will be referred to in this paper as a weak linear vortex.

Regarding the computational aspects of the resulting optimal control problem with constraints and inspired by the geometry behind this problem, we propose a modification of the optimization scheme previously considered in [Li-Xu-Teo-Chu, Time-optimal Zermelo's navigation problem with moving and fixed obstacles, 2013] by adding a piecewise constant rotation. This modification will entail adding another piecewise constant control to the problem which in turn proves to make the resulting approximated time-optimal paths more precise and efficient as we argue by the example of navigation through a linear vortex.

In this paper, we mainly prove that on Kenmotsu and Sasakian statistical manifolds, the Riemannian curvature tensor and the statistical curvature tensor fields are equal, only if their covariant derivatives are equal.

In this paper, we introduce multi-Dirac structures for Lie bialgebroids, which generalize the multi-Dirac structures on manifolds and Dirac structures on Lie bialgebroids. Next, we also introduce higher-order Courant algebroids for Lie algebroids and higher-order Dorfman algebroids for Lie algebroids and study the relationship between them. Furthermore, we show that there is a one-to-one correspondence between the multi-Dirac structures for special Lie bialgebroids and the higher Dirac structures for Lie algebroids. Finally, we construct the Gerstenhaber algebra by using the multi-Dirac structure for Lie bialgebroids.

The aim of this paper is to classify the connected Lie groups which act isometrically and with cohomogeneity c, where $c\in \{0,1\}$, on the de Sitter space $d{S}^{n-1,1}$ up to conjugacy in $SO(n,1)$ and then up to orbit equivalence. Among other results, we give the list of the groups represented in the isometry group of the de Sitter space $d{S}^{n-1,1}$.

In this paper, we examine the modal aspects of higher groups in Shulman's Cohesive Homotopy Type Theory. We show that every higher group sits within a modal fracture hexagon which renders it into its discrete, infinitesimal, and contractible components. This gives an unstable and synthetic construction of Schreiber's differential cohomology hexagon. As an example of this modal fracture hexagon, we recover the character diagram characterizing ordinary differential cohomology by its relation to its underlying integral cohomology and differential form data, although there is a subtle obstruction to generalizing the usual hexagon to higher types. Assuming the existence of a long exact sequence of differential form classifiers, we construct the classifiers for circle k-gerbes with connection and describe their modal fracture hexagon.

In this paper, we focus on a conformally flat Riemannian manifold $(M^n,g)$ of dimension n isometrically immersed into the $(n+1)$-dimensional light-cone ${\Lambda }^{n+1}$ as a hypersurface. We compute the first and the second variational formulas on the volume of such hypersurfaces. Such a hypersurface $M^n$ is not only immersed in ${\Lambda }^{n+1}$ but also isometrically realized as a hypersurface of a certain null hypersurface $N^{n+1}$ in the Minkowski spacetime, which is different from ${\Lambda }^{n+1}$. Moreover, $M^n$ has a volume-maximizing property in $N^{n+1}$.

The purpose of this paper is to determine a locally conformal Kähler solvmanifold such that its associated solvable Lie group is a one-dimensional extension of a 2-step nilpotent Lie group.

We introduce pseudo-spherical non-null framed curves in the three-dimensional anti-de Sitter spacetime and establish the existence and uniqueness of these curves. We then give moving frames along pseudo-spherical framed curves, which are well-defined even at singular points of the curve. These moving frames enable us to define evolutes and focal surfaces of pseudo-spherical framed immersions. We investigate the singularity properties of these evolutes and focal surfaces. We then reveal that the evolute of a pseudo-spherical framed immersion is the set of singular points of its focal surface. We also interpret evolutes and focal surfaces as the discriminant and the secondary discriminant sets of certain height functions, which allows us to explain evolutes and focal surfaces as wavefronts from the viewpoint of Legendrian singularity theory. Examples are provided to flesh out our results, and we use the hyperbolic Hopf map to visualize these examples.